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Autoreducibility of Random Sets: A Sharp Bound on the Density of Guessed Bits

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Book cover Mathematical Foundations of Computer Science 2002 (MFCS 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2420))

Abstract

A set A ⊆ 0,1* is called i.o. Turing-autoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don’t-know symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truth-table autoreducible. Ebert and Vollmer obtained the somewhat counterintuitive result that every Martin-Löf random set is i.o. truth-table-autoreducible and investigated the question of how dense the set of guessed bits can be when i.o. autoreducing a random set.

We show that rec-random sets are never i.o. truth-table-autoreducible such that the set of guessed bits has strictly positive constant density in the limit, and that a similar assertion holds for Martin-Löf random sets and i.o. Turing-autoreducibility. On the other hand, our main result asserts that for any computable function r that goes non-ascendingly to zero, any rec-random set is i.o. truth-table-autoreducible such that the set of guessed bits has density bounded from below by r(m).

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Author information

Authors and Affiliations

  1. Department of Computer Engineering and Computer Science, California State University, Long Beach, 1250 Bellflower Blvd., Long Beach, Ca, 90840, USA

    Todd Ebert

  2. Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 294, D-69120, Heidelberg, Germany

    Wolfgang Merkle

Authors
  1. Todd Ebert
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  2. Wolfgang Merkle
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Editor information

Editors and Affiliations

  1. Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097, Warszawa, Poland

    Krzysztof Diks  & Wojciech Rytter  & 

  2. Department of Computer Science, Liverpool University, Peach Street, Liverpool, L69 7ZF, UK

    Wojciech Rytter

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Ebert, T., Merkle, W. (2002). Autoreducibility of Random Sets: A Sharp Bound on the Density of Guessed Bits. In: Diks, K., Rytter, W. (eds) Mathematical Foundations of Computer Science 2002. MFCS 2002. Lecture Notes in Computer Science, vol 2420. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45687-2_18

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  • DOI: https://doi.org/10.1007/3-540-45687-2_18

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  • Print ISBN: 978-3-540-44040-6

  • Online ISBN: 978-3-540-45687-2

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